I think only thing challenging on these difficulties is early game, if you can survive the early game any civ game, you good to go.

Haven't played much of these civ games, but I remember playing the toughest difficulty the AI starts with 2 settlers, an army, and I think they aggressive. Thats it.

If you looking for a smart AI, clever tricky etc. I don't think any civ game when you up the difficulty the civ AI gets tougher.
If anything they get cheaper, get tech cheap get units cheap, etc. The AI being smarter isn't a thing.

So if you get good position end game, you right doesn't matter the difficulty, its easy.
Harder AI is just tough survival game until you are #1 and you control the balance of the other AIs.

I did find some logic where if you don't meet a civ or bunch of civs aka the other contient of civs, they fight each other more and can keep getting bigger and bigger to push you for the #1 spot.

Only advice I can say if you like tough and challenging times throughout your game is, let the AI goto war, let some AI takeover and get powerful. I'm not sure how you play but if you keep everyone in check, so no AI can be pretty powerful, you not going to experience a late game tough time against the AI.

I made a mod that should make the game more difficult by changing some of the rules.
In particular it removes a bunch of game-mechanics that the AI has trouble with and introduced some workarounds to fix stuff like AI not building certain essential buildings.

You can download it here:
https://github.com/Xilmi/Civ3AILove/releases/download/AIL_v_8_1/AIL_V8_1.BIQ

Let me know if you can still beat the game on the highest difficulty.

It is also advised to use the "C3X" inofficial patch as it fixes some bugs that also affected the AI.
https://forums.civfanatics.com/resources/c3x.28759/

If you just want something more challenging already in game, and you are open to playing some of the scenarios in the Game, Teturkitan Cities (where everyone starts with all the cities founded) is highly unbalanced in a way that can make playing certain civs (with fewer or sub-optimal cities) highly challenging.

You could, in addition to what others have mentioned, look into rolling back the patch that blocks the AI from building Armies. The game reportedly becomes rather nightmarish with that patch undone on higher difficulties.